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Stream: learning: questions

Topic: differentiation/integration

Scott Lingran (Sep 17 2020 at 13:32):

What's the category theoretic understanding of differentiation and integration in calculus?

Right now I understand them as 2-morphisms in the category of Set, which transforms functions into other functions and is isomorphic. Would love any thoughts or if you're able to point me to any materials.

John Baez (Sep 17 2020 at 13:57):

There are no 2-morphisms in the category $\mathsf{Set}$, since $\mathsf{Set}$ is a not a 2-category.

I think you're saying something like this: if $D(\mathbb{R})$ is the set of differentiable functions $f: \mathbb{R} \to \mathbb{R}$ and $\mathbb{R}^\mathbb{R}$ is the set of all functions $f : \mathbb{R} \to \mathbb{R}$, then there is a function

$\displaystyle{ \frac{d}{dx} : D(\mathbb{R}) \to \mathbb{R}^\mathbb{R} }$

sending any function to its derivative.

John Baez (Sep 17 2020 at 13:58):

There's a similar statement for integration. There are many ways to make these statements more complicated and powerful, which are studied in the subject called real analysis.

John Baez (Sep 17 2020 at 13:58):

There are also other ways to think about derivatives and integrals using more category theory.

Nathanael Arkor (Sep 17 2020 at 14:04):

You might be interested in the work on Cartesian differential categories. The category of smooth maps $\mathbb R^n \to \mathbb R^m$ is a motivating example.

James Wood (Sep 17 2020 at 14:58):

Did these authors ever cover integration? I feel as if I have a decent abstract understanding of differentiation, but that understanding is perhaps in conflict with integration.

Nathanael Arkor (Sep 17 2020 at 15:02):

Cockett and Lemay have a paper called Cartesian Integral Categories and Contextual Integral Categories.

Nathanael Arkor (Sep 17 2020 at 15:02):

(There's a talk here, too.)

Paolo Perrone (Sep 17 2020 at 15:56):

The derivative can be seen as a functor, in that sense it maps functions to functions.
I've explained this a little bit in my notes here, Example 1.3.25. (I'm sure there are more specific references on this, though.)
I don't know about integration of functions. Anyone? (There are measure monads, but that's a bit different.)

JS PL (he/him) (Sep 18 2020 at 10:51):

James Wood said:

Did these authors ever cover integration? I feel as if I have a decent abstract understanding of differentiation, but that understanding is perhaps in conflict with integration.

I've worked on the integral side of the story. Thanks @Nathanael Arkor for sharing my paper with Robin. You might also be interested in these papers:
https://arxiv.org/abs/1707.08211, the first paper on integral categories
https://arxiv.org/pdf/1902.04555.pdf, where we give an integral category structure from integrating smooth functions
I also have some notes on "Cartesian differential categories with antiderivatives", if you're interested.

JS PL (he/him) (Sep 18 2020 at 10:52):

But the integral story is far from being finished! Still lots to do.

JS PL (he/him) (Sep 18 2020 at 10:55):

Scott Lee said:

What's the category theoretic understanding of differentiation and integration in calculus?

As @Nathanael Arkor pointed out, you might be interested in differential categories. Last summer I gave a talk introducing the first levels of differential categories:
https://pages.cpsc.ucalgary.ca/%7Erobin/FMCS/FMCS2019/slides/Simon-LemaySlidesFMCS2019.pdf
Always happy to talk differential categories! There are others on this board who work on differential categories like @Ben MacAdam, @Jonathan Gallagher , @Geoff Cruttwell , etc.

JR Learnstomath (Jan 20 2025 at 12:29):

John Baez said:

There are also other ways to think about derivatives and integrals using more category theory.

and @JS PL (he/him) mentioned two approaches to talking about differentiation in category theoretic terms, if I understand well.

Does this mean that there are multiple ways to talk about differentiation and integration using cateogry theory? Does someone have a way to talk about these compositionaly in order to teach differentiation and integration to, say advanced high school or undergrad math students?

Also, I'm interested in those notes on "Cartesian differential categories with antiderivatives", please @JS PL (he/him) . I'd love to talk about this more, but I just don't know enough to have a sensible conversation (yet!).

JR Learnstomath (Jan 20 2025 at 12:34):

Also, do these existing approaches cover both analytical and numeric/approximation solution-finding?

Jean-Baptiste Vienney (Jan 20 2025 at 17:57):

In my humble opinion, differential categories, although they are very cool, have not been developed a lot in the directions of concrete calculations and/or solving differential equations although I know that Marie Kerjean dreams to solve (partial) differential equations using the cut-elimination (an algorithm) of differential linear logic which is the logic/computer-science side of differential categories.

On my side, I hope that the tool that I call functional differential ring could help to solve (ordinary) differential, functional and functional differential equations maybe by extending differential Galois theory to a functional differential Galois theory of functional differential fields, but first I must learn differential Galois theory and first Galois theory. Hopefully, I'm taking a course on Galois theory this semester :)

John Baez (Jan 20 2025 at 18:24):

JR Learnstomath said:

Does this mean that there are multiple ways to talk about differentiation and integration using category theory?

All I know is that there are multiple ways to generalize differentiation to different contexts.

Does someone have a way to talk about these compositionally in order to teach differentiation and integration to, say advanced high school or undergrad math students?

I'm not aware of work in category theory that would help students learn calculus. Some approaches simply list axioms taken from calculus and use them to define more general calculus-like structures.

Also, do these existing approaches cover both analytical and numeric/approximation solution-finding?

Not that I've seen.

Spencer Breiner (Jan 20 2025 at 19:04):

Not related to differential categories, etc. but there is this paper using string diagrams for vector calculus:
https://arxiv.org/abs/1911.00892

John Baez (Jan 20 2025 at 19:12):

The most interesting part of that to me is equation (18), the relation between the vector cross product and dot product in 3 dimensions, which was emphasized by Penrose.

JS PL (he/him) (Jan 20 2025 at 20:25):

Jean-Baptiste Vienney said:

In my humble opinion, differential categories, although they are very cool, have not been developed a lot in the directions of concrete calculations and/or solving differential equations

I would disagree. People have looked at differential equations in tangent categories (and Cartesian differential categories). Though I agree that story is far from complete, or necessarily practical

JS PL (he/him) (Jan 20 2025 at 20:27):

John Baez said:

JR Learnstomath said:

Does this mean that there are multiple ways to talk about differentiation and integration using category theory?

All I know is that there are multiple ways to generalize differentiation to different contexts.

Does someone have a way to talk about these compositionally in order to teach differentiation and integration to, say advanced high school or undergrad math students?

I'm not aware of work in category theory that would help students learn calculus. Some approaches simply list axioms taken from calculus and use them to define more general calculus-like structures.

I agree with both of @John Baez view points here

JS PL (he/him) (Jan 20 2025 at 20:29):

JR Learnstomath said:

Also, I'm interested in those notes on "Cartesian differential categories with antiderivatives", please JS PL (he/him) . I'd love to talk about this more, but I just don't know enough to have a sensible conversation (yet!).

Yes I'm always happy to talk about differential categories! Happy to share some more recent slides about introductions to differential categories. (they seem that have picked a lot of interest and popularity in 2024).

Regarding "Cartesian differential categories with antiderivatives", Robin and I are in the process of writing up the paper version. My objective is to have a preprint up this year.

JR Pasmo (Jan 22 2025 at 05:37):

Very cool and very interesting, thank you! Lots for me to read up on and digest.

When Twitter was still Twitter, a prof said something to the effect of, "differentiation is [left?] adjoint to integration, think about it", and I have been, intermittently. Unfortunately, I now can't find the exact post...

JR Pasmo (Jan 22 2025 at 06:09):

Jean-Baptiste Vienney said:

...I know that Marie Kerjean dreams to solve (partial) differential equations using the cut-elimination (an algorithm) of differential linear logic which is the logic/computer-science side of differential categories.

Is this the cut-elimination algorithm you refer to?

Yes, hopefully you can take that course on Galois theory!

Jean-Baptiste Vienney (Jan 22 2025 at 12:57):

The paper you linked contains the cut-elimination for a different logic. The logic its cut elimination that I was referring to are described in this paper.

Jean-Baptiste Vienney (Jan 22 2025 at 12:58):

I’m already taking the course on Galois theory in fact! (I think I didn’t express myself clearly.)

Jean-Baptiste Vienney (Jan 22 2025 at 13:01):

I have taken a look quickly as an introduction to differential Galois theory and they say that some things are weird because there isn’t a composition operation in differential fields so I think my project could really make sense to get a slightly better theory which can prove that some functional differential equations don’t have a solution which can be expressed in terms of a choice of elementary functions . At least, I hope so!

Jean-Baptiste Vienney (Jan 22 2025 at 13:05):

Anyway, that’s a good excuse to learn differential Galois theory.

Jean-Baptiste Vienney (Jan 22 2025 at 13:24):

(It will probably not work :sweat_smile: )

JR Learnstomath (Jan 26 2025 at 17:18):

Jean-Baptiste Vienney said:

The paper you linked contains the cut-elimination for a different logic. The logic its cut elimination that I was referring to are described in this paper.

Ah, okay, thank you!

JR Learnstomath (Jan 26 2025 at 17:23):

Thanks for sharing, all still a bit over my head right now, but really good to know the kinds of things I can be focusing on.

Jean-Baptiste Vienney said:

(It will probably not work :sweat_smile: )

And even if it doesn't work, you'll probably learn a lot from the process that will make something work eventually??

Jean-Baptiste Vienney (Jan 26 2025 at 17:40):

JR Learnstomath said:

Thanks for sharing, all still a bit over my head right now, but really good to know the kinds of things I can be focusing on.

Jean-Baptiste Vienney said:

(It will probably not work :sweat_smile: )

And even if it doesn't work, you'll probably learn a lot from the process that will make something work eventually??

Yes! I want to learn about linear algebraic groups now because they say it's a prerequisite for differential Galois theory :sweat_smile: